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Properties

Label 1200.3.c.e.449.4
Level $1200$
Weight $3$
Character 1200.449
Analytic conductor $32.698$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1200,3,Mod(449,1200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1200.449"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,-136] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.4
Root \(1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 1200.449
Dual form 1200.3.c.e.449.3

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.23607 + 2.00000i) q^{3} -2.00000i q^{7} +(1.00000 + 8.94427i) q^{9} -13.4164i q^{11} -8.00000i q^{13} -13.4164 q^{17} -34.0000 q^{19} +(4.00000 - 4.47214i) q^{21} -40.2492 q^{23} +(-15.6525 + 22.0000i) q^{27} +40.2492i q^{29} -14.0000 q^{31} +(26.8328 - 30.0000i) q^{33} +56.0000i q^{37} +(16.0000 - 17.8885i) q^{39} -26.8328i q^{41} +8.00000i q^{43} -40.2492 q^{47} +45.0000 q^{49} +(-30.0000 - 26.8328i) q^{51} -40.2492 q^{53} +(-76.0263 - 68.0000i) q^{57} +13.4164i q^{59} -46.0000 q^{61} +(17.8885 - 2.00000i) q^{63} -32.0000i q^{67} +(-90.0000 - 80.4984i) q^{69} -53.6656i q^{71} +106.000i q^{73} -26.8328 q^{77} -22.0000 q^{79} +(-79.0000 + 17.8885i) q^{81} +120.748 q^{83} +(-80.4984 + 90.0000i) q^{87} -107.331i q^{89} -16.0000 q^{91} +(-31.3050 - 28.0000i) q^{93} +122.000i q^{97} +(120.000 - 13.4164i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{9} - 136 q^{19} + 16 q^{21} - 56 q^{31} + 64 q^{39} + 180 q^{49} - 120 q^{51} - 184 q^{61} - 360 q^{69} - 88 q^{79} - 316 q^{81} - 64 q^{91} + 480 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.23607 + 2.00000i 0.745356 + 0.666667i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.00000i − 0.285714i −0.989743 − 0.142857i \(-0.954371\pi\)
0.989743 − 0.142857i \(-0.0456289\pi\)
\(8\) 0 0
\(9\) 1.00000 + 8.94427i 0.111111 + 0.993808i
\(10\) 0 0
\(11\) − 13.4164i − 1.21967i −0.792527 − 0.609837i \(-0.791235\pi\)
0.792527 − 0.609837i \(-0.208765\pi\)
\(12\) 0 0
\(13\) − 8.00000i − 0.615385i −0.951486 − 0.307692i \(-0.900443\pi\)
0.951486 − 0.307692i \(-0.0995567\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −13.4164 −0.789200 −0.394600 − 0.918853i \(-0.629117\pi\)
−0.394600 + 0.918853i \(0.629117\pi\)
\(18\) 0 0
\(19\) −34.0000 −1.78947 −0.894737 − 0.446594i \(-0.852637\pi\)
−0.894737 + 0.446594i \(0.852637\pi\)
\(20\) 0 0
\(21\) 4.00000 − 4.47214i 0.190476 − 0.212959i
\(22\) 0 0
\(23\) −40.2492 −1.74997 −0.874983 − 0.484153i \(-0.839128\pi\)
−0.874983 + 0.484153i \(0.839128\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −15.6525 + 22.0000i −0.579721 + 0.814815i
\(28\) 0 0
\(29\) 40.2492i 1.38790i 0.720021 + 0.693952i \(0.244132\pi\)
−0.720021 + 0.693952i \(0.755868\pi\)
\(30\) 0 0
\(31\) −14.0000 −0.451613 −0.225806 − 0.974172i \(-0.572502\pi\)
−0.225806 + 0.974172i \(0.572502\pi\)
\(32\) 0 0
\(33\) 26.8328 − 30.0000i 0.813116 − 0.909091i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 56.0000i 1.51351i 0.653697 + 0.756757i \(0.273217\pi\)
−0.653697 + 0.756757i \(0.726783\pi\)
\(38\) 0 0
\(39\) 16.0000 − 17.8885i 0.410256 − 0.458681i
\(40\) 0 0
\(41\) − 26.8328i − 0.654459i −0.944945 − 0.327229i \(-0.893885\pi\)
0.944945 − 0.327229i \(-0.106115\pi\)
\(42\) 0 0
\(43\) 8.00000i 0.186047i 0.995664 + 0.0930233i \(0.0296531\pi\)
−0.995664 + 0.0930233i \(0.970347\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −40.2492 −0.856366 −0.428183 − 0.903692i \(-0.640846\pi\)
−0.428183 + 0.903692i \(0.640846\pi\)
\(48\) 0 0
\(49\) 45.0000 0.918367
\(50\) 0 0
\(51\) −30.0000 − 26.8328i −0.588235 − 0.526134i
\(52\) 0 0
\(53\) −40.2492 −0.759419 −0.379710 − 0.925106i \(-0.623976\pi\)
−0.379710 + 0.925106i \(0.623976\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −76.0263 − 68.0000i −1.33379 − 1.19298i
\(58\) 0 0
\(59\) 13.4164i 0.227397i 0.993515 + 0.113698i \(0.0362697\pi\)
−0.993515 + 0.113698i \(0.963730\pi\)
\(60\) 0 0
\(61\) −46.0000 −0.754098 −0.377049 − 0.926193i \(-0.623061\pi\)
−0.377049 + 0.926193i \(0.623061\pi\)
\(62\) 0 0
\(63\) 17.8885 − 2.00000i 0.283945 − 0.0317460i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 32.0000i − 0.477612i −0.971067 − 0.238806i \(-0.923244\pi\)
0.971067 − 0.238806i \(-0.0767560\pi\)
\(68\) 0 0
\(69\) −90.0000 − 80.4984i −1.30435 − 1.16664i
\(70\) 0 0
\(71\) − 53.6656i − 0.755854i −0.925835 − 0.377927i \(-0.876637\pi\)
0.925835 − 0.377927i \(-0.123363\pi\)
\(72\) 0 0
\(73\) 106.000i 1.45205i 0.687666 + 0.726027i \(0.258635\pi\)
−0.687666 + 0.726027i \(0.741365\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −26.8328 −0.348478
\(78\) 0 0
\(79\) −22.0000 −0.278481 −0.139241 − 0.990259i \(-0.544466\pi\)
−0.139241 + 0.990259i \(0.544466\pi\)
\(80\) 0 0
\(81\) −79.0000 + 17.8885i −0.975309 + 0.220846i
\(82\) 0 0
\(83\) 120.748 1.45479 0.727396 − 0.686218i \(-0.240731\pi\)
0.727396 + 0.686218i \(0.240731\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −80.4984 + 90.0000i −0.925270 + 1.03448i
\(88\) 0 0
\(89\) − 107.331i − 1.20597i −0.797753 − 0.602985i \(-0.793978\pi\)
0.797753 − 0.602985i \(-0.206022\pi\)
\(90\) 0 0
\(91\) −16.0000 −0.175824
\(92\) 0 0
\(93\) −31.3050 − 28.0000i −0.336612 − 0.301075i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 122.000i 1.25773i 0.777514 + 0.628866i \(0.216481\pi\)
−0.777514 + 0.628866i \(0.783519\pi\)
\(98\) 0 0
\(99\) 120.000 − 13.4164i 1.21212 − 0.135519i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1200.3.c.e.449.4 4
3.2 odd 2 inner 1200.3.c.e.449.2 4
4.3 odd 2 300.3.b.c.149.1 4
5.2 odd 4 1200.3.l.r.401.1 2
5.3 odd 4 240.3.l.a.161.2 2
5.4 even 2 inner 1200.3.c.e.449.1 4
12.11 even 2 300.3.b.c.149.3 4
15.2 even 4 1200.3.l.r.401.2 2
15.8 even 4 240.3.l.a.161.1 2
15.14 odd 2 inner 1200.3.c.e.449.3 4
20.3 even 4 60.3.g.a.41.1 ✓ 2
20.7 even 4 300.3.g.d.101.2 2
20.19 odd 2 300.3.b.c.149.4 4
40.3 even 4 960.3.l.a.641.2 2
40.13 odd 4 960.3.l.d.641.1 2
60.23 odd 4 60.3.g.a.41.2 yes 2
60.47 odd 4 300.3.g.d.101.1 2
60.59 even 2 300.3.b.c.149.2 4
120.53 even 4 960.3.l.d.641.2 2
120.83 odd 4 960.3.l.a.641.1 2
180.23 odd 12 1620.3.o.b.1241.2 4
180.43 even 12 1620.3.o.b.701.2 4
180.83 odd 12 1620.3.o.b.701.1 4
180.103 even 12 1620.3.o.b.1241.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.3.g.a.41.1 ✓ 2 20.3 even 4
60.3.g.a.41.2 yes 2 60.23 odd 4
240.3.l.a.161.1 2 15.8 even 4
240.3.l.a.161.2 2 5.3 odd 4
300.3.b.c.149.1 4 4.3 odd 2
300.3.b.c.149.2 4 60.59 even 2
300.3.b.c.149.3 4 12.11 even 2
300.3.b.c.149.4 4 20.19 odd 2
300.3.g.d.101.1 2 60.47 odd 4
300.3.g.d.101.2 2 20.7 even 4
960.3.l.a.641.1 2 120.83 odd 4
960.3.l.a.641.2 2 40.3 even 4
960.3.l.d.641.1 2 40.13 odd 4
960.3.l.d.641.2 2 120.53 even 4
1200.3.c.e.449.1 4 5.4 even 2 inner
1200.3.c.e.449.2 4 3.2 odd 2 inner
1200.3.c.e.449.3 4 15.14 odd 2 inner
1200.3.c.e.449.4 4 1.1 even 1 trivial
1200.3.l.r.401.1 2 5.2 odd 4
1200.3.l.r.401.2 2 15.2 even 4
1620.3.o.b.701.1 4 180.83 odd 12
1620.3.o.b.701.2 4 180.43 even 12
1620.3.o.b.1241.1 4 180.103 even 12
1620.3.o.b.1241.2 4 180.23 odd 12